I. Introduction

From time-to-time, one has the opportunity to assist in developing a research tool with the versatility of a Swiss Army Knife (or sonic screwdriver for the Dr. Who aficionados). I was lucky enough to have that opportunity while working on my postdoc at the University of Louisville Comparative Planetology Laboratory (CPL) http://louisville.edu/cpl/ established and lead by Prof. Timothy Dowling in 2000. The primary goal of CPL is to develop theoretical and numerical models that may be used to gain a better understanding of the mechanisms that shape large-scale planetary atmosphere characteristics.

Early astronomers first noted changes in appearance of planets such as Venus, Mars, Jupiter, Saturn, Uranus and Neptune as well as their larger moons. As the optical resolution of their instruments increased, they were able to deduce that these planetary bodies were, in fact, enshrouded by gaseous envelopes with features not dissimilar to those noted in Earth atmosphere including belts, zones, and vortices. With the advent of planetary probe programs such as Mariner, Venera, Magellan, Pioneer, Voyager, Galileo, and Cassini, enough remotely sensed and in-situ data was collected from most of the planets and larger moons to ascertain that large variations exist from one planetary body to the next including large-scale flows analogous to jet streams found in Earth's atmosphere to the size, duration, rotational direction and speed of cyclonic storms.

However, despite these difference, hydrostatic primitive equations can be used to model large-scale flows in both terrestrial and gas giant planetary atmospheres. With sufficient flexibility allowed in the forcing terms that drive the conservation of momentum, thermal energy, and continuity equations which comprise the hydrostatic primitive equations, it should be possible to develop a single model that may be adapted to conditions specific to multiple planetary atmospheres without investing the time and computational resources in developing specialized models with applicability to just a single planetary atmosphere. The simplicity of this approach provides the motivation for the development of the Explicit Planetary Isentropic Coordinate Model (EPIC) (LeBeau and Dowling 1998; Dowling et al. 2006). "Explicit" relates to the time step employed as well as the finite-differencing scheme; "Planetary" merely indicates that the model is designed to employ physics and dynamics parameterizations consistent with conditions observed in the atmospheres of several different planets and moons; and "Isentropic Coordinate" refers to the vertical coordinate framework employed in the model. It is this last component, the Isentropic Coordinate that lends EPIC its unique modeling flexibility.

II. Hybrid Vertical Coodinate


Fig. 1  (Dowling et al. 2006) diverges from for the case of Jupiter and Earth.

The EPIC vertical coordinate scheme is based on a hybrid system developed by Konor and Arakawa (1997) in which coordinate variable is prescribed using the following formulation:

where ranges from 0 at the bottom of the model atmosphere and 1 at the top of the model atmosphere and is potential temperature. In this formulation, the boundary values of  and are prescribed such that:

where is a free parameter corresponding to our hybrid coordinate at the model bottom. To insure that remain monotonic in the vertical direction, a number of assumptions are made in Konor and Arakawa (1997) including the slope of the gradient with respect to . From these assumptions a final expression for our hybrid coordinate can be derived such that:


where and are minimum model potential temperature and potential temperature vertical gradient, respectively.

This hybrid vertical coordinate affords several advantages over traditional -based isentropic or -based scaled pressure coordinate systems including:

1. Reduction in the tendency for numerical instabilities to develop as a result of model levels "pinching together" when isentropic coordinates are used near terrain features such as mountains and plateaus associated with terrestrial planets such as Venus, Earth, and Mars. To remedy this situation, one must either reformulate the model level intervals near the lower boundary or reduce the model timestep. Neither of these approaches is satisfactory, however, since they compromise the model's ability to yield useful information about physics and dynamics processes near the model bottom.

2. Utility in modeling the interior of gas giant planets which possesses an abyssal layer characterized by near adiabatic conditions and zonal winds that are a function of only latitude. Little dynamic evolution occurs at the abyssal layer so that it may serve as a proxy for the solid surfaces associated with terrestrial planets.

3. A reduction in the computational expense associated with simulating vertical momentum and heat fluxes. In the upper model layers, isentropic behavior dominates so that adiabatic process do not induce vertical motions within the model. Rather only diabatic processes such as radiative heating and cooling can induce vertical motion. This significantly reduces the number of calls to dynamics routines needed during the model integration.

Fig. 1 (Dowling et al. 2006) depicts how typical profiles of and derived from our formulation for Earth and Jupiter. It illustrates how the hybrid coordinate mimics an isentropic framework aloft yet diverges near the solid surface (in the case of Earth) or near the abyssal layer (in the case of Jupiter).  To gain a better appreciation of how this hybrid coordinate would appear in physical space, contours of constant are interpolated onto pressure levels and plotted above an exaggerated representation of venusian terrain derived from the Magellan synthetic aperture radar sensor data record in Fig. 2 and 3 (Colon and Dowling 2000). Of note is the terrain-following nature of near the surface.


Fig. 2 Zonal cross-section of hybrid coordinate         Fig. 3 Oblique view of the hybrid coordinate
overlying exaggerated venusian terrain.               overlying exaggerated venusian terrain.

III. Numerical Validation

A critical requirement for evaluating a new general circulation model's ability to realistically represent large scale atmospheric flows is to be able to accurately reproduce results derived in a number of established test cases. One of these was developed by Held and Suarez (1994) and has served as a benchmark for all subsequent numerical model validation studies. This test uses simple Newtonian cooling and Rayleigh drag parameterizations to simulate large-scale circulation patterns observed in Earth's atmosphere but may be adapted to model circulation features in any terrestrial or gas giant planet.  The EPIC model test specifics are discussed in detail in Dowling et al. (2006) and summarized as follows:

• 24 vertical model layers employed spanning pressure levels ranging from 1000 mbar to 1 mbar.
• Arakawa C grid horizontal grid discretization.
• horizontal resolution.
• =260K; pure coordinate corresponds to a pressure level of 900 mbar and below.
• Timestep = 150 s; 3rd-order Adams-Bashforth timestep employed for the momentum equations.
• Adjustments to hyperviscosity and divergence damping parameters are scaled relative to the Adams-Bashforth timestep.
• US Standard Atmosphere is used for the initial temperature profile.
• Run initialized with zero wind and no perturbations.
• To initiate the dynamical evolution of the run, a small perturbation is added in the form of two weak anticyclones with amplitudes of 1 m/s are "switched on" within the first 200 days of model simulation time.
• The model is run for 200 days to allow for dynamical spin-up and statistics are then recorded from 200 to 1200 days with a sampling rate of 2 days.
• The resulting diagnostic fields are interpolated onto scaled pressure coordinates relative to surface pressure i.e. p/psurf.

Fig. 4 (Dowling et al. 2006) EPIC model results for the Earth-atmosphere benchmark of Held and Suarez (1994). The statistics are generated by extracting variables from the run at a sampling rate of 2 days. (a) Mean zonal wind [m/s ]; (b) mean of the square of the temperature eddies [ ]; (c) mean temperature [K]; (d) mean potential temperature [K], with the contours in 5 K intervals except 325, 350, 400, and 500 K.

Fig. 4 (a) depicts the zonal wind averaged over time and longitude. Midlatitude jets form as expected and the zonal-wind structure reproduce the original paper’s results in detail, which is significant given the difference in vertical coordinates between EPIC and the original paper's pure -based vertical coordinate. Fig. 4(b) shows the squared eddy temperature averaged over time and longitude. It matches the original as well as the zonal-wind plot. Fig 4(c) and 4(d) illustrate the mean temperature and potential temperature isotherms averaged over time and longitude which, like the plots of the other diagnostic variables, match the results in the original paper for the most part except for a higher the expected temperature gradient near the model tropics. Qualitatively, in this brief description of test results, the EPIC model is able to reproduce realistic large-scale circulation patterns. A quantitative analysis is covered in detail in Dowling et al. (2006).

Summary

In this brief description of the EPIC general circulation model, a novel approach in specifying vertical coordinates is described with emphasis on its flexibility in handling atmospheric conditions that may be encountered on terrestrial planets and moons or gas giant worlds. Starting with the hybrid coordinate formulation of Konor and Arakawa (1997), a new hybrid coordinate was adapted to suit our needs based on preserving the utility of a pure isentropic coordinate while still being able to handle its shortcomings near terrestrial planet terrain features and gas giant abyssal layers. A number of studies have already utilized the EPIC model to examine phenomena ranging from the modeling of cyclonic storms in Jupiter's atmosphere (Morales-Juberias et al. 2003) to the impact of terrain features on Venus's large-scale circulation. (Hernnstein and Dowling 2007). Future work will focus on the continued modeling large-scale atmospheric circulation features observed on other worlds including Saturn's large moon, Titan, from which a great deal of in-situ and remotely sensed data was collected during the Cassini-Huygens mission.

References
 
Colon, E., and T.E. Dowling, 2000, EPIC simulations of topographic effects on the general circulation model of Venus, Bull. Amer. Astron. Soc. 32, 873-879.

Dowling, T.E., Bradley, M.E., Colon, E., Kramer, J.,LeBeau, R.P., Lee, G.C.H., Mattox, T.I., Morales-Juberías, R., Palotai C.s.J., Parimi, V.K., and A. P. Showman, 2006, The EPIC atmospheric model with an isentropic/terrain-following hybrid vertical coordinate, Icarus 182, 259-273.

Held, I.M., and M. J. Suarez, 1994. A proposal for the intercomparison of the dynamical cores of atmospheric general circulation models. Bull. Am. Meteorol. Soc. 73, 1825–1830.

Konor, C.S., and A. Arakawa, 1997. Design of an atmospheric model based on a generalized vertical coordinate. Mon. Weather Rev. 125, 1649–1673.

LeBeau, R.P. and T.E. Dowling, 1998, EPIC simulations of time-dependent, three dimensional vortices with application to Neptune's Great Dark Spot, Icarus 132: 239-265.

Morales-Juberias, R., Sanchez-Lavega, A., and T.E. Dowling, 2003, EPIC simulations of the merger of Jupiter’s White Ovals BE and FA: Altitude-dependent behavior, Icarus 166, 63-74.